ORDINAL1: The Ordinal Numbers. Transfinite Induction and Defining by Transfinite Induction

:: The Ordinal Numbers. Transfinite Induction and Defining by Transfinite Induction
:: by Grzegorz Bancerek
::
:: Received March 20, 1989
:: Copyright (c) 1990-2012 Association of Mizar Users

begin

theorem :: ORDINAL1:1

canceled;

theorem :: ORDINAL1:2

canceled;

theorem :: ORDINAL1:3

canceled;

theorem :: ORDINAL1:4

canceled;

theorem Th5: :: ORDINAL1:5

for Y, X being set st Y in X holds
not X c= Y

proof end;

definition

let X be set ;

func succ X -> set equals :: ORDINAL1:def 1
X \/ {X};

coherence ;

end;

:: deftheorem defines succ ORDINAL1:def 1 :

registration

let X be set ;

cluster succ X -> non empty ;

coherence ;

end;

theorem Th6: :: ORDINAL1:6

for X being set holds X in succ X

proof end;

theorem :: ORDINAL1:7

for X, Y being set st succ X = succ Y holds
X = Y

proof end;

theorem Th8: :: ORDINAL1:8

for x, X being set holds
( x in succ X iff ( x in X or x = X ) )

proof end;

theorem Th9: :: ORDINAL1:9

for X being set holds X <> succ X

proof end;

definition

let X be set ;

attr X is epsilon-transitive means :Def2: :: ORDINAL1:def 2
for x being set st x in X holds
x c= X;

attr X is epsilon-connected means :Def3: :: ORDINAL1:def 3
for x, y being set st x in X & y in X & not x in y & not x = y holds
y in x;

end;

:: deftheorem Def2 defines epsilon-transitive ORDINAL1:def 2 :

:: deftheorem Def3 defines epsilon-connected ORDINAL1:def 3 :

Lm1: ( {} is epsilon-transitive & {} is epsilon-connected )

proof end;

::
:: 4. Definition of ordinal numbers - Ordinal
::

definition

let IT be set ;

attr IT is ordinal means :Def4: :: ORDINAL1:def 4
( IT is epsilon-transitive & IT is epsilon-connected );

end;

:: deftheorem Def4 defines ordinal ORDINAL1:def 4 :

registration

cluster ordinal -> epsilon-transitive epsilon-connected for number ;

coherence by Def4;

cluster epsilon-transitive epsilon-connected -> ordinal for number ;

coherence by Def4;

end;

notation

synonym number for set ;

end;

registration

cluster ordinal for number ;

existence by Lm1;

end;

definition

mode Ordinal is ordinal number ;

end;

theorem Th10: :: ORDINAL1:10

for A, B being set
for C being epsilon-transitive set st A in B & B in C holds
A in C

proof end;

theorem Th11: :: ORDINAL1:11

for x being epsilon-transitive set
for A being Ordinal st x c< A holds
x in A

proof end;

theorem :: ORDINAL1:12

for A being epsilon-transitive set
for B, C being Ordinal st A c= B & B in C holds
A in C

proof end;

theorem Th13: :: ORDINAL1:13

for a being set
for A being Ordinal st a in A holds
a is Ordinal

proof end;

theorem Th14: :: ORDINAL1:14

for A, B being Ordinal holds
( A in B or A = B or B in A )

proof end;

definition

let A, B be Ordinal;

:: original: c=
redefine pred A c= B means :: ORDINAL1:def 5
for C being Ordinal st C in A holds
C in B;

compatibility

proof end;

connectedness

proof end;

end;

:: deftheorem defines c= ORDINAL1:def 5 :

theorem :: ORDINAL1:15

for A, B being Ordinal holds A,B are_c=-comparable

proof end;

theorem Th16: :: ORDINAL1:16

for A, B being Ordinal holds
( A c= B or B in A )

proof end;

registration

cluster empty -> ordinal for number ;

coherence by Lm1;

end;

theorem Th17: :: ORDINAL1:17

for x being set st x is Ordinal holds
succ x is Ordinal

proof end;

theorem Th18: :: ORDINAL1:18

for x being set st x is ordinal holds
union x is ordinal

proof end;

registration

cluster non empty epsilon-transitive epsilon-connected ordinal for number ;

existence

proof end;

end;

registration

let A be Ordinal;

cluster succ A -> non empty ordinal ;

coherence by Th17;

cluster union A -> ordinal ;

coherence by Th18;

end;

theorem Th19: :: ORDINAL1:19

for X being set st ( for x being set st x in X holds
( x is Ordinal & x c= X ) ) holds
X is ordinal

proof end;

theorem Th20: :: ORDINAL1:20

for X being set
for A being Ordinal st X c= A & X <> {} holds
ex C being Ordinal st
( C in X & ( for B being Ordinal st B in X holds
C c= B ) )

proof end;

theorem Th21: :: ORDINAL1:21

for A, B being Ordinal holds
( A in B iff succ A c= B )

proof end;

theorem Th22: :: ORDINAL1:22

for A, C being Ordinal holds
( A in succ C iff A c= C )

proof end;

::
:: 6. Transfinite induction and principle of minimum of ordinals
::

scheme :: ORDINAL1:sch 1
OrdinalMin{ P₁[ Ordinal] } :

ex A being Ordinal st
( P₁[A] & ( for B being Ordinal st P₁[B] holds
A c= B ) )

provided

A1: ex A being Ordinal st P₁[A]

proof end;

Transfinite induction

scheme :: ORDINAL1:sch 2
TransfiniteInd{ P₁[ Ordinal] } :

for A being Ordinal holds P₁[A]

provided

A1: for A being Ordinal st ( for C being Ordinal st C in A holds
P₁[C] ) holds
P₁[A]

proof end;

::
:: 7. Properties of sets of ordinals
::

theorem Th23: :: ORDINAL1:23

for X being set st ( for a being set st a in X holds
a is Ordinal ) holds
union X is ordinal

proof end;

theorem Th24: :: ORDINAL1:24

for X being set st ( for a being set st a in X holds
a is Ordinal ) holds
ex A being Ordinal st X c= A

proof end;

theorem Th25: :: ORDINAL1:25

for X being set holds
not for x being set holds
( x in X iff x is Ordinal )

proof end;

theorem Th26: :: ORDINAL1:26

for X being set holds
not for A being Ordinal holds A in X

proof end;

theorem :: ORDINAL1:27

for X being set ex A being Ordinal st
( not A in X & ( for B being Ordinal st not B in X holds
A c= B ) )

proof end;

::
:: 8. Limit ordinals
::

definition

let A be set ;

attr A is limit_ordinal means :Def6: :: ORDINAL1:def 6
A = union A;

end;

:: deftheorem Def6 defines limit_ordinal ORDINAL1:def 6 :

theorem Th28: :: ORDINAL1:28

for A being Ordinal holds
( A is limit_ordinal iff for C being Ordinal st C in A holds
succ C in A )

proof end;

theorem :: ORDINAL1:29

for A being Ordinal holds
( not A is limit_ordinal iff ex B being Ordinal st A = succ B )

proof end;

::
:: 9. Transfinite sequences
::

definition

let IT be set ;

attr IT is T-Sequence-like means :Def7: :: ORDINAL1:def 7
proj1 IT is ordinal ;

end;

:: deftheorem Def7 defines T-Sequence-like ORDINAL1:def 7 :

registration

cluster empty -> T-Sequence-like for number ;

coherence

proof end;

end;

definition

mode T-Sequence is T-Sequence-like Function;

end;

registration

let Z be set ;

cluster Relation-like Z -valued Function-like T-Sequence-like for number ;

existence

proof end;

end;

definition

let Z be set ;
mode T-Sequence of Z is Z -valued T-Sequence;

end;

theorem :: ORDINAL1:30

for Z being set holds {} is T-Sequence of Z

proof end;

theorem :: ORDINAL1:31

for F being Function st dom F is Ordinal holds
F is T-Sequence of rng F by Def7, RELAT_1:def 19;

registration

let L be T-Sequence;

cluster proj1 L -> ordinal ;

coherence by Def7;

end;

theorem :: ORDINAL1:32

for X, Y being set st X c= Y holds
for L being T-Sequence of X holds L is T-Sequence of Y

proof end;

registration

let L be T-Sequence;
let A be Ordinal;

cluster L | A -> rng L -valued T-Sequence-like ;

coherence

proof end;

end;

theorem :: ORDINAL1:33

for X being set
for L being T-Sequence of X
for A being Ordinal holds L | A is T-Sequence of X ;

definition

let IT be set ;

attr IT is c=-linear means :Def8: :: ORDINAL1:def 8
for x, y being set st x in IT & y in IT holds
x,y are_c=-comparable ;

end;

:: deftheorem Def8 defines c=-linear ORDINAL1:def 8 :

theorem :: ORDINAL1:34

for X being set st ( for a being set st a in X holds
a is T-Sequence ) & X is c=-linear holds
union X is T-Sequence

proof end;

::
:: 10. Schemes of definability by transfinite induction
::

scheme :: ORDINAL1:sch 3
TSUniq{ F₁() -> Ordinal, F₂( T-Sequence) -> set , F₃() -> T-Sequence, F₄() -> T-Sequence } :

F₃() = F₄()

provided

A1: ( dom F₃() = F₁() & ( for B being Ordinal
for L being T-Sequence st B in F₁() & L = F₃() | B holds
F₃() . B = F₂(L) ) ) and
A2: ( dom F₄() = F₁() & ( for B being Ordinal
for L being T-Sequence st B in F₁() & L = F₄() | B holds
F₄() . B = F₂(L) ) )

proof end;

scheme :: ORDINAL1:sch 4
TSExist{ F₁() -> Ordinal, F₂( T-Sequence) -> set } :

ex L being T-Sequence st
( dom L = F₁() & ( for B being Ordinal
for L1 being T-Sequence st B in F₁() & L1 = L | B holds
L . B = F₂(L1) ) )

proof end;

scheme :: ORDINAL1:sch 5
FuncTS{ F₁() -> T-Sequence, F₂( Ordinal) -> set , F₃( T-Sequence) -> set } :

for B being Ordinal st B in dom F₁() holds
F₁() . B = F₃((F₁() | B))

provided

A1: for A being Ordinal
for a being set holds
( a = F₂(A) iff ex L being T-Sequence st
( a = F₃(L) & dom L = A & ( for B being Ordinal st B in A holds
L . B = F₃((L | B)) ) ) ) and
A2: for A being Ordinal st A in dom F₁() holds
F₁() . A = F₂(A)

proof end;

theorem :: ORDINAL1:35

for A, B being Ordinal holds
( A c< B or A = B or B c< A )

proof end;

begin

:: moved from ORDINAL2, 2006.06.22, A.T.

definition

let X be set ;

func On X -> set means :Def9: :: ORDINAL1:def 9
for x being set holds
( x in it iff ( x in X & x is Ordinal ) );

existence

proof end;

uniqueness

proof end;

func Lim X -> set means :: ORDINAL1:def 10
for x being set holds
( x in it iff ( x in X & ex A being Ordinal st
( x = A & A is limit_ordinal ) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines On ORDINAL1:def 9 :

:: deftheorem defines Lim ORDINAL1:def 10 :

Generalized Axiom of Infinity

theorem Th36: :: ORDINAL1:36

for D being Ordinal ex A being Ordinal st
( D in A & A is limit_ordinal )

proof end;

definition

func omega -> set means :Def11: :: ORDINAL1:def 11
( {} in it & it is limit_ordinal & it is ordinal & ( for A being Ordinal st {} in A & A is limit_ordinal holds
it c= A ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def11 defines omega ORDINAL1:def 11 :

registration

cluster omega -> non empty ordinal ;

coherence by Def11;

end;

definition

let A be set ;

attr A is natural means :Def12: :: ORDINAL1:def 12
A in omega ;

end;

:: deftheorem Def12 defines natural ORDINAL1:def 12 :

registration

cluster natural for number ;

existence

proof end;

end;

definition

mode Nat is natural number ;

end;

registration

natural number ex b₁ being number st
for b₂ being natural number holds b₂ in b₁

proof end;

end;

:: from ARYTM_3, 2006.05.26

registration

let A be Ordinal;

cluster -> ordinal for Element of A;

coherence

proof end;

end;

:: missing, 2006.06.25, A.T.

registration

cluster natural -> ordinal for number ;

coherence

proof end;

end;

:: from ZF_REFLE, 2007,03.13, A.T.

scheme :: ORDINAL1:sch 6
ALFA{ F₁() -> non empty set , P₁[ set , set ] } :

ex F being Function st
( dom F = F₁() & ( for d being Element of F₁() ex A being Ordinal st
( A = F . d & P₁[d,A] & ( for B being Ordinal st P₁[d,B] holds
A c= B ) ) ) )

provided

A1: for d being Element of F₁() ex A being Ordinal st P₁[d,A]

proof end;

:: from CARD_4, 2007.08.06, A.T.

theorem :: ORDINAL1:37

for X being set holds (succ X) \ {X} = X

proof end;

:: from ARYTM_3, 2007.09.16, A.T.

registration

cluster empty -> natural for number ;